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In [[linear algebra]], an '''orthogonal diagonalization''' of a symmetric [[matrix (mathematics)|matrix]] is a [[diagonalizable matrix|diagonalization]] by means of an [[orthogonal matrix|orthogonal]] change of coordinates.
 
The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form ''q''(''x'') on '''R'''<sup>''n''</sup> by means of an orthogonal change of coordinates ''X''&nbsp;=&nbsp;''PY''.<ref>Lipschutz, Seymour. ''3000 Solved Problems in Linear Algebra.''</ref>
 
* Step 1: find the [[symmetric matrix]] A which represents q and find its [[characteristic polynomial]] <math>\Delta (t).</math>
* Step 2: find the [[eigenvalues]] of A which are the [[Root system|roots]] of <math>\Delta (t)</math>.
* Step 3: for each eigenvalues <math>\lambda</math> of A in step 2, find an orthogonal basis of its [[eigenspace]].
* Step 4: normalize all eigenvectors in step 3 which then form an orthonormal basis of '''R'''<sup>''n''</sup>.
* Step 5: let P be the matrix whose columns are the normalized [[eigenvector]]s in step 4.
The X=PY is the required orthogonal change of coordinates, and the diagonal entries of <math>P^T AP</math> will be the eigenvalues <math>\lambda_{1} ,\dots ,\lambda_{n}</math> which correspond to the columns of P.
 
==References==
{{reflist}}
 
[[Category:Linear algebra]]
 
{{linear-algebra-stub}}

Revision as of 22:03, 20 March 2013

In linear algebra, an orthogonal diagonalization of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates.

The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on Rn by means of an orthogonal change of coordinates X = PY.[1]

The X=PY is the required orthogonal change of coordinates, and the diagonal entries of PTAP will be the eigenvalues λ1,,λn which correspond to the columns of P.

References

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Template:Linear-algebra-stub

  1. Lipschutz, Seymour. 3000 Solved Problems in Linear Algebra.